Fluid Mechanics

When numerical methods are combined with fluid mechanics, you ultimately get Computational Fluid Dynamics (CFD.) CFD has been a test of numerical methods for a long time, especially with the solution of the Navier-Stokes equations. The non-linear nature of these equations has made mating them with CFD a challenging problem.

When you’re first introduced to numerical methods you’re told that there are ways one can predict the stability and consistency of these methods with a given set of physics. These methods are helpful but things go wrong. On top of that coding these solutions into a language like C++ or Fortran for parallel processing (or any other computer application) is a challenging business; the endless parade of bugs we see exploited in internet security can be also seen in modelling physics such as fluid flow, just different bugs. Probably the most important lesson one learns in mastering such an art is that, just because it comes out of a computer, it can’t always be trusted.

That situation confronted a Turkish friend of mine who was working on a particularly sticky CFD coding problem. One frustrating day he exclaimed, “I need to find the way,” i.e., out of his CFD coding problem.

Finding the way has been a key issue in the Middle East for a long time. The last night before he was crucified, Jesus Christ had this exchange with his disciples:

“We do not know where you are going, Master,” said Thomas; “so how can we know the way?”

Jesus answered: “I am the Way, and the Truth, and the Life; no one ever comes to the Father except through me. (John 14:5-6 TCNT)

This, then, is a key difference between Jesus Christ and the others. They knew (or they thought they knew) the way, or they have an idea of what the way is. But Jesus Christ is the way, something he showed when he rose from the dead shortly thereafter.

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In our last post we discussed the basics of hydraulic jump. We showed how the flow across the jump could be estimated using the depths of the water before and after the jump. In this post we will show another method, using a pitot tube, which is also used in a wide variety of other applications.

The calculations in our earlier post assumed the velocity of the water to be uniform. In a flow channel with parallel straight sides and rectangular cross-section, we assumed the flow to be

where the variables are as before. For both open and closed channels, however, the velocity will not be uniform; the velocity we used was an average velocity. If we actually measured the velocity at various points in the flow, we would get different results for different places, as illustrated below (from Open Channel Flow Measurement):

The following presentation assumes a rectangular cross-section with parallel walls.

Pitot Tube Measurements

We discussed the concept behind the Pitot tube in Wind Tunnel Testing. The simplest way to show this for open channel flow is to use the presentation of Vennard (1940):

Integrating the Results

Separating the static pressure from the stagnation (dynamic) pressure is fairly simple in open channel flow, although reading the results can be tricky if the flow level varies. We thus take several measurements at several heights and compute a velocity at each point, getting a velocity profile for the flow. But how to integrate this relationship for a total flow? There are several methods.

One would be to take a trend line of the results (as discussed in Least Squares Analysis and Curve Fitting) and integrate it in closed form. For this to work requires that the trend line have a very good correlation with the original data.

Another would be to take the results and graphically integrate them. For many engineering applications, graphical methods were popular for many years because they avoided many difficult calculations which strained the limited computational power of the time. Although the results were approximate, with CAD a higher degree of precision can be obtained than was possible before. The tricky part of a graphical method such as this is the scaling, which must be understood to properly interpret the results.

Yet another is the use of numerical integration, generally piecewise with methods such as the Trapezoidal Rule and Simpson’s Rule. With a properly laid out spreadsheet, this can be done with minimal effort, although attention to detail is crucial to success.

An implementation of numerical integration can be used which simplifies the calculations, and is suggested in Open Channel Flow Measurement. It involves a little pre-planning in that the points where the data is taken need to be pre-determined (they should be in any case.) Since the more points of data the more accurate the result (all other things equal,) we’ll use six points. Those points are at the surface, at the bottom of the channel, and at 20% (0.2), 40% (0.4), 60% (0.6) and 80% (0.8) of the total depth. The mean velocity can be thus computed as follows:

Boundary layer considerations would indicate that the velocity at the bottom of the channel be zero, but if it is possible to take a measurement it would be better.

Once the is calculated, it can be used in the equation at the beginning of the post to estimate the flow in the channel.

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Hydraulic jump is one of the more interesting phenomena in open channel flow. It is used, for example, in stilling basins, where it is necessary to reduce the fluid velocity coming out from a dam by dissipating the kinetic energy in the flow. An example of what it looks like is below.

This is a very basic treatment of the subject. It is based on Streeter (1966) and also Dodge and Thompson (1937), where many of the graphics come from. To keep things simple we will assume that the jump takes place in an open channel with two vertical parallel walls. But first we need to introduce a couple of concepts.

Although it’s confusing, the Froude Number is frequently the square root of this value, and for this study will be designated by the equation

In this case is the average velocity of the fluid in a one-dimensional channel; the fact that it’s average is very important. The constant is the gravitational constant and is the depth of the fluid (usually water) in the channel. This is the way we will use the Froude Number in this study.

Since we’re restricting ourselves to vertically-walled channels with rectangular cross-sections, we can make another modification to the Froude Number. If we define the flow as

where is the flow in volume per unit time, is the width of the channel and is defined as before, the Froude Number can be stated as

Keep in mind this is only for channels with vertical walls and rectangular flow cross-sections.

Specific Energy

Let us consider a vertical cross section of a one-dimensional horizontal channel flow. Bernoulli’s Equation states that the fluid energy/head at that cross section should be

What we have done here is to eliminate the pressure term in Bernoulli’s Equation, which does not apply for a surface. We are left with the potential term (the depth of the fluid in the channel) and the kinetic term (due to the movement of the fluid.) The energy is thus in units of length, or is a head term.

Again assuming vertical, parallel walls and substituting the flow term, this equation can be stated as

The resulting curve looks like this:

Obviously this figure allows for non-parallel sides, but the basic theory is the same. The depth which we call is called in this diagram. At a depth of zero the specific energy is infinite, decreasing to a minimum at the point where the curve in (c) comes closest to the ordinate, and then decreases to the point where the second term on the right hand side vanishes and the specific energy equals to the depth. We can find the location of this minimum by differentiating with respect to depth as follows:

Solving this equation for yields a cubic equation with one real and two complex roots. Using the real root only, the depth at which the energy is minimised is given by the equation

This is referred to as the critical depth, and flow at this point is referred to as critical flow. For depths shallower than this, the flow is referred to as rapid or supercritical. For depths above this, the flow is referred to as subcritical or tranquil.

Some algebra will also reveal the following:

At critical flow, the Froude Number is unity.

At critical flow, the specific energy is .

Since the Froude Number is inversely proportional to , for values of y greater than critical (subcritical/tranquil flow) the Froude Number is less than unity.

Conversely, for values of y less than critical (supercritical/rapid flow) the Froude Number is greater than unity.

Solving for Hydraulic Jump

Now that we’ve laid the groundwork for some basics of flow in channels, it’s time to consider hydraulic jump itself, which is illustrated below.

The first thing worth noting is that the incoming flow (from the left) must be supercritical for hydraulic jump to take place. From that the flow on the right must be subcritical. In the process of the jump energy is dissipated, which is a big purpose of inducing hydraulic jump in, say, a stilling basin. But this also means that conservation of energy does not apply here. What does apply is a) conservation of momentum and b) conservation of mass flow.

Starting with conservation of momentum, after a little algebra and using our y notation for depth (as opposed to d in the figures) we have

After a great deal more algebra and application of the Froude Number,

Conservation of mass flow requires that

or

There is more than one way to combine the conservation of momentum and mass flow equations. One way results in determining the relationship between the two Froude Numbers, thus

Plot of the Froude Number before the jump (Fr1) and after (Fr2)

We can also reapply the conservation of mass flow equation and eliminate in this way:

We now have a definite relationship between a) the two Froude Numbers and b) the ratio of the water depths on both sides of the jump to the first Froude Number (and thus the second.)

Determining the Flow

At this point we have the basics of hydraulic jump. There are many ways we can apply this theory. In this monograph we’ll look at two: using the results to estimate the flow, and using the results to estimate the energy loss.

Let’s consider a worked example, taken from Streeter (1966). A hydraulic jump takes place where the upstream depth is 5′ and the downstream depth is 31′. The sluice is 50′ wide. Determine the Froude Numbers on either side of the jump and the flow through the jump in cubic feet per second.

We start by rewriting the equation that relates the ratio of depths to the upstream Froude Number (1) as follows:

Solving for ,

Substituting, .

We then substitute this result into this equation

and compute .

Since we now know the Froude Numbers for both sides, we can compute the flow. We defined the Froude Number earlier in terms of flow; rearranging that equation yields

Taking either the upstream Froude Number and depth or the downstream Froude Number and depth and substituting these, the sluice width (50′) and the gravitational constant, the result .

Computing the Energy Loss

One of the main purposes for inducing hydraulic jump is to dissipate kinetic energy in a fluid flow. Earlier we saw that at any point in the flow there is a specific energy, or head, in the flow. If we subtract the specific energy at the upstream (Point 1) from the downstream (Point 2) the difference between the two heads is

or strictly a function of the two water depths. If we substitute the two water depths from the last example, this yields an energy/head change of 28.35′. Although the units are of length, it is more accurate to say that this is an energy per unit weight of water, or ft-lb/lb. The power dissipated during the hydraulic jump is thus

Substituting the head change we just computed with the flow we computed earlier and the unit weight of water yields P = 26,510,196 ft-lb/sec = 48,200 hp.

You are urged to set these equations up in a spreadsheet and substitute the values given to confirm the validity of your spreadsheet before using them for another case.

References (other than ones on this site)

Dodge, R.A., and Thompson, M.J. (1937) Fluid Mechanics. First Edition. New York: McGraw-Hill Book Company

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In the earlier post on buoyancy and stability, methods for the determination of the metastatic height are discussed. This is yet another method of doing this, taken (with a few modifications) from E.H. Lewitt, Hydraulics and Fluid Mechanics (Sir Isaac Pitman and Sons, 1923.) This text also has a treatment on the “moment of inertia” method of computing metastatic height.

The metastatic height of a ship or pontoon may be found experimentally whilst the
vessel is floating, if the position of the centre of gravity is known.

Let W be the weight of the ship, which is known, and let G be the centre of gravity. Let a known movable weight m be placed on one side of the ship. A pendulum consisting of a weight suspended by a long cord is placed in the ship, and the position of the bob when at rest is marked. Let l be the length of the pendulum. The weight m is then moved across the deck through the distance x, the new position of m being denoted by m’. This will cause the ship to swing through a small angle about its metacentre M. Then, as the pendulum inside the ship still remains vertical, the angle may be measured by the apparent deflection of the pendulum.

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James N. Warrington
Associate, American Society of Naval Engineers
Presented at the Annual Meeting, 22 December 1893
Reprinted in the Naval Engineers Journal, July 1983

Since the investigations of the elder Froude, friction and wave making have been fully recognized as the leading elements in the resistance of ships. The resistance due to eddies is known to be quite inconsiderable in modern ships, while the variations in resistance which accompany change in trim, bodily rise or subsidence of the vessel, and wave interference. all occur at such speeds as produce waves of great magnitude relative to the displacement. Within the limit of speed at which these phenomena appear, friction and wave making may be regarded as supplying substantially the entire resistance, and it is here essayed to formulate the propulsive power absorbed by these two elements of resistance with with such a degree of approximation as may be attained by attributing the properties of a trochoidal wave to the waves of a ship.

Friction

The horse-power of the frictional resistance may be expressed by the formula

in which represents the area of the wetted surface, the speed in knots, and a constant to be derived from the coefficient of friction. For a clean painted surface exceeding fifty feet on length Froude’s experiments show a resistance of 0.25 lb. per square foot at a speed of ten feet per second, and 1.83 as the power of the speed to which the resistance is proportional. From these quantities are decided the values and , the speed being expressed in knots of 6,080 feet and the wetted surface in square feet.

Wave Making

USS Cincinnati, New York, NY, June 1896.

Considering the wave generated by the ship in motion, it may be observed that similar ships produce similar waves, and that for a given ship the ratio of the magnitude of any series of waves to the magnitude of any other series, as for example of the bow waves to the stern waves, will be constant at all speeds so long as their generation is free from mutual interference. Moreover the wave of any series will retain their similarity at all speeds, for the orbital velocity as well as the speed of advance of the wave form must both be proportional to the speed of the ship.

The aggregate energy of all the wave of the ship may therefore be conceived to be embodied in a single series of representative waves having a speed of advance equal to the speed of the ship.

As the particles of water in the longitudinal stream lines approach the ship they meet the component stream lines diverging from the apex of the bow wave and follow the resultant stream lines around the ship. The velocity in the diverging stream lines must clearly be sufficient to impart the necessary transverse displacement during the time occupied in the passage from the bow to the midship section. The diverging stream-line velocity will then be proportional to , in which is the speed of the ship, is the area of the midship section, and the length of the combined entrance and run. But this diverging stream-line velocity is closely proportional to, if not identical with, the orbital velocity of the bow waves. Hence these orbital velocities will also be proportional to , and since the wave of each series grow in like proportion with increase of speed, the orbital velocity of the representative wave also will be proportional to the same ratio.

Assuming now that the waves of the ship possess the properties of trochoidal waves, it follows that the height of the representative wave will be proportional to , in which represents the wave length, the orbital velocity, and the speed of advance. Furthermore, the breadth of the wave will be proportional to its height, for as the height increases the orbital velocity increases throughout the entire breadth of the wave, and imparts its motion to more remote particles previously at rest.

The horse-power of a series of trochoidal waves, having a length , a height , and a breadth , is thus expressed by Mr. Albert W. Stahl, U.S. Navy:

The ratio is constant for similar ships and for all speeds of a given ship, and is proportional to . Since with such a range in this latter ratio as may be found between a torpedo boat and an armored cruiser, the quantity in parenthesis would vary by a fraction of one percent, it may reasonably be neglected. The wave horse-power then becomes proportional to . But is proportional to , and h to , hence the horse-power absorbed in wave making may be expressed by the formula

in which is a constant to be derived from the performance of a model ship.

USS Montgomery, Tompkinsville (Staten Island), NY, June 1896

The imaginary representative wave is conceived as having a speed of advance equal to the speed of the ship, and in this respect is plainly in harmony with the transverse series. The speed of the advance of the diverging waves is dependent not only upon the speed of the ship but also upon the angle of divergence of their crest lines, and this angle of divergence is dependent upon the bluntness of the ship, or in other words upon the ratio , so long as the mode of distribution of the displacement remains the same. The length of the diverging waves will then vary with the ratio, and the effect will be that in the blunter ship the diverging series will absorb a greater amount of energy relative to the transverse series than in the ships of finer form. This relative increase in energy in the diverging series must, however, be accompanied by a corresponding diminution of the energy in the transverse series, for the total energy in both series is derived from the diverging component stream lines, the velocity of which is proportional to , and this total energy thus derived will be independent of the relative importance of the two waves. The condition necessary to the preceding statement is that the model ship, whence the constant is derived, and the projected ship shall be similar in cross section and identical in mode of distribution of the displacement. When this condition is fulfilled, therefore, the representative wave becomes a true measure of all the waves subject to the approximation involved in its trochoidal nature, and in the modification expression (2).

Concerning the limit of speed in the application of the formula, it should be observed that each wave-making feature is assumed to generate its appropriate wave without interference; hence the beginning of each interference marks the limit of its applicability.

At corresponding speeds the power absorbed in wave making becomes proportional to the square root of the seventh power of the scale of comparison, as should be expected.

The Wake Current

The sum of and is the effective horse-power required to tow the ship. When propulsion is effected by a screw propeller at the stern, other varying elements are introduced. The following current, by delivering momentum to the propeller, restores a portion of the energy already expended in overcoming the frictional and wave-making resistances.

The Pathfinder

Let represent the speed of the ship, the mean speed of the following current, and the diameter of the propeller. Then the energy imparted to the propeller per unit of time by the following current will be proportional to

but assuming a constant percentage of real slip, is proportional to

which being substituted in (4) reduces the increment of energy to the form

The following current has three components of independent origin, viz: the frictional wake, the orbital velocity of the stern wave, and the orbital velocity of the recurring transverse bow wave. It has been shown that when the waves of the bow series interfere with the formation of the stern wave the limit to the application of this analysis has been reached. Therefore the effect of the bow wave need not be considered.

The speed of the frictional wake will here be regarded as proportional to the speed of the ship; hence the energy derived from it, being a constant proportion of the total power, need not be considered here.

Still attributing the properties of the trochoidal wave to the waves of the ship, the orbital velocity of the stern wave at the center of the propeller will be proportional to

in which is the length of the after boddy, the modulus of the hyperbolic system of logarithms, the depth of immersion of the axis of the propeller, and a constant such that equals the wave length.

Substituting (6) for in expression (5) the increment of energy becomes

The value of constant as derived from the results of the experiment of Mr. George A. Calvert is ; but this value should be regarded as provisional, since like depends upon the mode of distribution of the displacement.

Summary

Let represent the total effective horse-power required at the propeller. Then

and are constants, depending upon the form of the ship, having a provisional value of 1.66.

Analysis of Power

If from the indicated horsepower, the power required to overcome the constant resistance of the unloaded engine is deducted, the result will be the new horse-power applied to the propulsion; the the ratio of to this net horse-power will represent the combined efficiency of the mechanism of the engine so far as the variable friction is involved, and of the propeller, including the effect of the frictional wake and of the augmentation of resistance by the propeller. These latter two effects are thus assumed proportional to the power applied. The ratio may then be expected to remain constant so long as the propeller efficiency remains constant.

Applications

U.S.S. Bancroft

The results of the progressive trial of the U.S.S. Bancroft given by Passed Assistant Engineer Robert S. Griffin, U.S. Navy, suffice for determination of the value of the constant of the vessel. The Bancroft is a twin-screw practice ship of the following dimensions: length of immersed body ft.; displacement on trial tons; area of immersed midship section on trial sq.ft.; wetted surface on trial sq.ft.

Having calculated the power absorbed by surface friction on the basis of Froude’s coefficient and exponent, and also the value of the factor involving the following current, the constant is given such a value as will maintain the ratio approximately constant. The value of thus found is for the Bancroft 329, and ratio . The resistance of the unloaded engine is assumed at 2 lbs. per square inch of the area of the low-pressure pistons. Using these values, the power at the several speeds hsa been calculated and is given in Table I, where the observed powers are also given show the degree of approximation.

In order to illustrate the scope of the formula the ratio is given in Table II together with explanatory data for each of seven vessels, five of them being twin-screw ships of the U.S. Navy, and the two others the S.S. Umbria of the Cunard line and the torpedo boat Sunderland, built by Messrs. William Doxford and Sons, Sunderland, England. With the exception of the Umbria these vessels may be regarded as approximations to the form of least resistance in smooth water. The Umbria has a quasi-parallel middle body, the length of which has been approximated by ascribing to the combined entrance and run a prismatic coefficient of fineness of 0.56. Table III contains the results of the progressive trial of the Umbria together with the calculated power. The same comparison is shown in Figure I.

In making these calculations, the constants of the Bancroft have been employed, their values being . If should be stated that the application of this value of to all of these vessels is not strictly warranted, for they are undoubtedly dissimilar in cross section and also in mode of distribution of displacement. Nevertheless it is assumed that they possess a sufficient degree of similarity to serve the purpose of illustration.

The resistance of the unloaded engine is based on two pounds per square inch of the area of the low-pressure pistons in all cases. The length is the combined length of entrance and run, being the length of the immersed body in all cases except the Umbria. The ratio given as an index to the character of the ship form a propulsive point of view. If all these vessels possessed the required degree of similarity, and had propellers and engines of equal mechanical efficiency, the ratio should be the same for all, provided the speeds were within the limit indicated.

Limit of Speed

Since the formula gives only the power absorbed in surface friction and in waves generated without interference, it is obvious that it may be used as a measure of the total power only when these elements constitute the entire resistance. The earliest departure from the condition of applicability may be expected to accompany an interference by the recurring transverse bow wave in the generation of the stern wave. Since the relative magnitude of the bow and stern waves is dependent upon the form of the ship, it is clearly impossible to formulate generally the limit at which such an interference will occur. The limit must therefore be found form the performance of the model ship whence the constants in any case are derived.

The U.S.S. Bancroft, within a speed of 14.52 knots, gives no evidence of the limit, as may be seen by reference to Table I. At this speed .

The S.S. Umbria gives no indication of the limit within the speed of 20 knots, as may be seen by Table III or Fig. I. At 20 knots length of ship.

On the other hand, the progressive trial of the torpedo boat Sunderland affords a convenient illustration of the inapplicability of the formula. Figure 2 shows the power curves, both actual and calculated.

At the speed of 14 knots the value of the ratio is such as to warrant the conclusion that at this speed the formula gives the power with sufficient accuracy. A glance at the curves, however, shows that this cannot be said of any higher speed until 20 knots is approached. Fourteen knots may then be regarded as the limit of speed at which the formula may be applied. At this speed . Beyond this speed the familar hump appears. It may be of interest to note that the curves intersect again in the region of 20 knots, at which .

It is not intended to question the existence of minor humps at speeds below the limits here noticed, but simply to observe that in the published curves of the Bancroft and Umbria they are not appreciable. It is possible that such humps may be partially neutralized by the action of a simultaneous increase in the wake current upon the propeller. In any event, their existence must affect the accuracy of the calculated power, although commonly to such as slight extent as to be inappreciable.

Since the magnitude of the waves of the ship varies with the ratio it follows that both the size of the hump and the speed at which it appears will be influenced by this ratio.

Conclusion

The method of this analysis may be summarized thus: The resistance of ships is so largely due to surface friction and wave making that all other forms may be neglected except at very high speeds. Within the limit of speed at which interference occurs in the generation of waves, the formula (eq. 8) enables the power absorbed by these two principal elements to be computed, provided the constants and have been experimentally determined by means of a model ship having the same mode of distribution of displacement as the projected ship. The ratio of the power thus computed to the net power of the engine is constant so long as the propeller efficiency is constant. Finally, the indicated horse power may be obtained by adding to the net power the power absorbed in the constant resistance of the engine.